Not Applicable
Not Applicable
Not Applicable
This invention relates to random number generators, specifically the use of cubic dice to generate random numbers from a predetermined set. Random number generators are used in commerce, industry, and education; for statistical sampling, simulation, and pure mathematical and scientific research. Statisticians have developed several basic sampling methods in order to obtain unbiased samples, that is, samples representative of the population. The most common methods are random, systematic, stratified, and cluster sampling. A sample is a subgroup of the population. By working with a sample rather than the entire population, researchers can save time and money, get more detailed information, and get information that would otherwise be impossible to obtain.
In random sampling, the basic requirement is that for a sample of size n, all possible samples of this size must have an equal chance of being selected. In order to meet this requirement, researchers can use one of two methods. The first method is to number each element of the population and then place the numbers on cards. Place the cards in container, mix them, and then select the sample by drawing the cards. Applicant""s invention improves upon this method by using cubic senary dice rather than numbered cards to choose the sample. The second way of selecting a random sample is to use random numbers. Random numbers can be obtained by using a calculator, computer, or a table. The theory behind random numbers is that each digit, 0 through 9, has an equal probability of occurring, so that in every ten digit sequence, each digit has a probability of 1/10 of occurring.
The first random number table, containing 41,600 digits, was published in 1927 by Cambridge University Press. The need for larger random number tables increased rapidlyxe2x80x94necessitated by very large sampling experiments. With the advent of digital computers, more and more random numbers were required for mathematical modeling and forecasting. Since computer memory was, at one time, too expensive to store random number tables, random numbers began to be generated by computers, programmed with random number algorithms.
These pseudo-random numbers, generated by computers, are statistically indistinguishable from genuine random numbers; but numbers generated by computer through a deterministic process cannot, by definition, be random. Given a knowledge of the algorithm used to create pseudo-random numbers and its internal state, it is possible to predict all the numbers returned by subsequent calls to the algorithm. The eminent mathematician John von Neumann stated in a 1951 paper: xe2x80x9cAnyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. For . . . there is no such thing as a random numberxe2x80x94there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.xe2x80x9d
Authentic random numbers, those numbers tending to occur with the same relative frequency in which a knowledge of one number or an arbitrarily long sequence of numbers is of no use in predicting the next number, can be generated by using mechanical devices. Probably the most familiar mechanical random number generators in the United States are those used for lotteries.
In a typical lottery random number generator, ten small plastic balls on which the decimal notational digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) have been printed, are placed in each of several small plexiglass containersxe2x80x94a separate container for the unit""s place, the ten""s place, the hundred""s place, etc. The plastic balls are energetically agitated and thoroughly mixed with compressed air. A ball is released from each container until a three or four digit number has been selected.
The drawing of the winning numbers is aboveboard and transparent to viewers. The public, both scientist and layperson alike, understands and trusts the random process by which the winning number is selected, and its inherent fairness. Confident public assurance of a lottery""s fair outcome, largely attributable to the mechanical random number generators used, is instrumental in garnering public support and participation in lotteries worldwide.
Since these devices are too expensive and cumbersome for most applications, except high stake lotteries, other types of mechanical random number generators have been developed, the simplest being dice. When cast, a pair of standard playing dice does not generate random numbers because the numbers from 2 to 12 tend to occur with different relative frequencies. However, random number generators using dice of various geometric shapes and markings have been patented. Unfortunately, most of these devices have deficiencies in terms of conceptual simplicity, fabrication cost, lack of consumer confidence regarding true randomness, user interface, and dynamic range.
Van Buskirk discloses in U.S. Pat. No. 6,158,738 a die for generating random numbers. Bowling discloses in U.S. Pat. No. 5,938,197 a random number generating die. Daniel et al. disclose in U.S. Pat. No. 5,909,874 a random number generator using icosahedron decimal dice. Sanditen discloses in U.S. Pat. No. 5,031,915 a random number generator using two elongated die. Crippen discloses in U.S. Pat. No. 4,497,487 a chance device comprising two icosahedrons. Palmer discloses in U.S. Pat. No. 4,239,226 a random number generator using ten-sided dice.
Cubic senary (base 6) dice each have numerical values uniquely corresponding to the place values of the base 6 number system, and face values uniquely corresponding to the senary notational digits (0, 1, 2, 3, 4, 5). The table below illustrates conceptually both the decimal and senary positional numeration systems:
Since the 6 faces of each cubic senary die are marked with the senary notational digits (0, 1, 2, 3, 4, 5). When cast, each die returns a numerical value from 0 to 5. When read from the highest to lowest power die, these notational digits generate a senary number. The decimal value of each die cast is the product of each die""s value (6 raised to its exponent""s value) multiplied by the notational digit""s intrinsic valuexe2x80x940, 1, 2, 3, 4, 5. The magnitude of the number cast is the sum of these products.
Three senary dice would generate senary numbers having 3 digits, ranging from 000 to 555, or from 0 to 215 when converted to decimal. For example, if the number 314 were to result when three cubic senary dice were cast, the 3 would indicate 3 thirty-sixes, the 1 would indicate 1 six, and the 4 would indicate 4 units (i.e. the decimal number 118), and not 3 hundreds, 1 ten and 4 units as in decimal numeration.
Cubic senary dice have several advantages over similar random number generating devices:
1. Cubic dice have been used for centuries by gamblers and mathematicians around the world. The familiar geometry of cubic dice is recognized and trusted to yield a fair outcome, that is, each of the six sides having an equal chance of turning up when cast. Cubic dice are easy to manufacture; and test for balance and symmetry. The relatively large face area on a cubic die allows for relatively large, easy to read, markings. For these reasons, in comparison with multi-faceted dice having more complex geometries, the traditional cube is clearly the winning configuration for random number generating dice.
2. Cubic senary dice are powerful. A handful of cubic senary dice can generate hundreds of thousands of random numbers. A cast of just seven cubic senary dice can generate any one of 67 or 279,936 random numbers, each of which will tend to occur with the same relative frequency. The number of cubic dice cast can easily be adjusted to accommodate user needs for a predetermined set of numbers. If the user wants to generate a million random numbers, eight dice would be cast. If only a thousand random numbers are needed then only four dice need be cast.
3. Generating random numbers with cubic senary dice is easy, and they are fun to use. The markings are clear, unambiguous, and easy to read. Concepts from the familiar decimal number system make the transition to the senary numbers easy. One quickly becomes accustomed to the base 6 number system when using cubic senary dice, giving one a sense of satisfaction and enjoyment from mastering a new and useful art.
In a preferred embodiment, for those who lack the inclination to reckon with senary numbers, cubic senary dice are marked with the familiar scientific notation format, so that it is unnecessary to work with senary (base 6) numbers when casting cubic senary dice. Also, in alternative embodiments, cubic senary die faces are marked with their decimal equivalents, and the cast evaluated by simple addition.